\section{积分学典型问题}
\subsection{不等式问题}
\begin{theorem}[Cauchy-Schwartz不等式]
    设\(f,g\)在\([a,b]\)上连续，则有
    \[
\left(\int_a^bf(x)g(x)\mathrm{d}x\right)^2\leq\int_a^bf^2(x)\mathrm{d}x\int_a^bg^2(x)\mathrm{d}x.
    \]
\end{theorem}
\begin{proof}
    因为
    \[0\leq \int_a^b(tf(x)+g(x))^2\mathrm{d}x=t^2\int_a^bf^2(x)\mathrm{d}x+2t\int_a^bf(x)g(x)\mathrm{d}x+\int_a^bg^2(x)\mathrm{d}x,
    \]
    所以
    \[
    \left(2\int_a^bf(x)g(x)
\mathrm{d}x\right)^2-4\int_a^bf^2(x)\mathrm{d}x\int_a^bg^2(x)\mathrm{d}x\leq 0,
    \]
    也就是
    \[
    \left(\int_a^bf(x)g(x)\operatorname{d}x\right)^2\leq\int_a^bf^2(x)\operatorname{d}x\int_a^bg^2(x)\operatorname{d}x.
    \]
\end{proof}

\begin{proposition}
    设$ f(x)$ 在$[a,b]$上三阶可导,存在 $\xi\in(a,b)$,使得
    \[
    f(b)=f(a)+\frac12(b-a)[f^{\prime}(a)+f^{\prime}(b)]-\frac1{12}(b-a)^3f^{\prime\prime\prime}(\xi).
    \]
\end{proposition}